Math RefereeA product counterexample to the simplex maximum
Projection-body volume and a Cartesian-product construction
The proposed argument
The argument computes projection-body volumes from facet vectors. An orthogonal Cartesian product makes the normalized volume multiplicative, while a simplex projection body is a scaled cube plus one diagonal segment. These identities reduce a product of two ten-dimensional simplices to an integer comparison with the twenty-dimensional simplex benchmark. Repeating ten-dimensional factors yields an exponential excess in all sufficiently large dimensions.
The main proposed counterexample and the growth corollary are separate claims. A human assessment has not yet been supplied; the reading materials and the panel commentary remain separate.
Reading order
- 01
Set the benchmark and scope
Define projection-body volume normalized by body volume, state the simplex benchmark and earlier results, and announce an explicit product witness and a consequence for large dimensions.
- 02
Obtain the geometric identities
Represent polytopal projection bodies by facet area-normal segments, separate the facets of an orthogonal Cartesian product, and track area-normal vectors under affine maps.
- 03
Compute the simplex benchmark
Write the standard simplex projection body as a scaled cube plus one diagonal segment; compute its volume by extending parallel fibers, then apply affine covariance to other simplices.
- 04
Compare products with simplices
Combine product multiplicativity and the simplex value for two ten-dimensional factors. Repeat such factors and add one simplex to reach every residue class, comparing their values to a fixed exponential upper estimate for the simplex benchmark.
Follow the dependencies
Open the full map ↗Select a claim to see its prerequisites and consumers. Edge labels distinguish proof inputs, source references, background and exclusions.
Your independent assessment
Reconstruct the facet formula, the Cartesian-product identity and the simplex calculation. Recompute the comparison in exact arithmetic. Check the corollary's quantifiers independently, and distinguish proof inputs from historical context.